3.2.1 \(\int (a+a \sin (c+d x))^{4/3} \, dx\) [101]

Optimal. Leaf size=67 \[ -\frac {2\ 2^{5/6} a \cos (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{d (1+\sin (c+d x))^{5/6}} \]

[Out]

-2*2^(5/6)*a*cos(d*x+c)*hypergeom([-5/6, 1/2],[3/2],1/2-1/2*sin(d*x+c))*(a+a*sin(d*x+c))^(1/3)/d/(1+sin(d*x+c)
)^(5/6)

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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2731, 2730} \begin {gather*} -\frac {2\ 2^{5/6} a \cos (c+d x) \sqrt [3]{a \sin (c+d x)+a} \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right )}{d (\sin (c+d x)+1)^{5/6}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[c + d*x])^(4/3),x]

[Out]

(-2*2^(5/6)*a*Cos[c + d*x]*Hypergeometric2F1[-5/6, 1/2, 3/2, (1 - Sin[c + d*x])/2]*(a + a*Sin[c + d*x])^(1/3))
/(d*(1 + Sin[c + d*x])^(5/6))

Rule 2730

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2^(n + 1/2))*a^(n - 1/2)*b*(Cos[c + d*x]/
(d*Sqrt[a + b*Sin[c + d*x]]))*Hypergeometric2F1[1/2, 1/2 - n, 3/2, (1/2)*(1 - b*(Sin[c + d*x]/a))], x] /; Free
Q[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 2731

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[a^IntPart[n]*((a + b*Sin[c + d*x])^FracPart
[n]/(1 + (b/a)*Sin[c + d*x])^FracPart[n]), Int[(1 + (b/a)*Sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int (a+a \sin (c+d x))^{4/3} \, dx &=\frac {\left (a \sqrt [3]{a+a \sin (c+d x)}\right ) \int (1+\sin (c+d x))^{4/3} \, dx}{\sqrt [3]{1+\sin (c+d x)}}\\ &=-\frac {2\ 2^{5/6} a \cos (c+d x) \, _2F_1\left (-\frac {5}{6},\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x))\right ) \sqrt [3]{a+a \sin (c+d x)}}{d (1+\sin (c+d x))^{5/6}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.16, size = 341, normalized size = 5.09 \begin {gather*} \frac {\left (-\frac {3}{2} (-5+\cos (c+d x)) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 (-1)^{3/4} e^{-\frac {3}{2} i (c+d x)} \left (i+e^{i (c+d x)}\right ) \left (20 e^{i (c+d x)} \sqrt {\cos ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )} \, _2F_1\left (-\frac {1}{3},\frac {1}{3};\frac {2}{3};-i e^{-i (c+d x)}\right )-2 \left (1+i e^{-i (c+d x)}\right )^{2/3} \left (1+e^{2 i (c+d x)}\right ) \, _2F_1\left (\frac {1}{2},\frac {5}{6};\frac {11}{6};\sin ^2\left (\frac {1}{4} (2 c+\pi +2 d x)\right )\right )+5 i \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};-i e^{-i (c+d x)}\right ) \sqrt {2-2 \sin (c+d x)}\right )}{4 \sqrt {2} \left (1+i e^{-i (c+d x)}\right )^{2/3} \sqrt {i e^{-i (c+d x)} \left (-i+e^{i (c+d x)}\right )^2}}\right ) (a (1+\sin (c+d x)))^{4/3}}{2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[c + d*x])^(4/3),x]

[Out]

(((-3*(-5 + Cos[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/2 + (3*(-1)^(3/4)*(I + E^(I*(c + d*x)))*(20*E
^(I*(c + d*x))*Sqrt[Cos[(2*c + Pi + 2*d*x)/4]^2]*Hypergeometric2F1[-1/3, 1/3, 2/3, (-I)/E^(I*(c + d*x))] - 2*(
1 + I/E^(I*(c + d*x)))^(2/3)*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1/2, 5/6, 11/6, Sin[(2*c + Pi + 2*d*x
)/4]^2] + (5*I)*Hypergeometric2F1[1/3, 2/3, 5/3, (-I)/E^(I*(c + d*x))]*Sqrt[2 - 2*Sin[c + d*x]]))/(4*Sqrt[2]*E
^(((3*I)/2)*(c + d*x))*(1 + I/E^(I*(c + d*x)))^(2/3)*Sqrt[(I*(-I + E^(I*(c + d*x)))^2)/E^(I*(c + d*x))]))*(a*(
1 + Sin[c + d*x]))^(4/3))/(2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3)

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (a +a \sin \left (d x +c \right )\right )^{\frac {4}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(d*x+c))^(4/3),x)

[Out]

int((a+a*sin(d*x+c))^(4/3),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(4/3),x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(4/3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(4/3),x, algorithm="fricas")

[Out]

integral((a*sin(d*x + c) + a)^(4/3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \sin {\left (c + d x \right )} + a\right )^{\frac {4}{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))**(4/3),x)

[Out]

Integral((a*sin(c + d*x) + a)**(4/3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(d*x+c))^(4/3),x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^(4/3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,\sin \left (c+d\,x\right )\right )}^{4/3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*sin(c + d*x))^(4/3),x)

[Out]

int((a + a*sin(c + d*x))^(4/3), x)

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